The gradient tells you how fast something is changing, and the area tells you how much has built up. Read the vertical axis and the question word before you pick a calculation.
This is the last lesson in the SPM Mathematics motion graphs chapter. It builds on calculating distance from a speed-time graph.
What decision table should you use?
| Graph | Question asks for | Use | Unit |
|---|---|---|---|
| Distance-time | Speed | Gradient | m/s |
| Distance-time | Distance from start | Read the height | m |
| Speed-time | Acceleration or deceleration | Gradient | m/s² |
| Speed-time | Distance travelled | Area under the line | m |
Two steps make the choice: name the vertical axis, then underline the quantity in the question.
Worked example: one graph, three questions
A speed-time graph has points (0, 0), (6, 12) and (10, 12), with speed in m/s and time in seconds.
Question 1: What is the acceleration in the first 6 seconds? Acceleration is a rate of change of speed, so use the gradient: (12 − 0) ÷ 6 = 2 m/s².
Question 2: How far does it travel in the first 6 seconds? Distance needs the area under the line. The shape is a triangle: ½ × 6 × 12 = 36 m.
Question 3: What is the total distance in 10 seconds? Add the rectangle from 6 to 10 s: 4 × 12 = 48. Then 36 + 48 = 84 m.
The mistake that costs marks
The common slip is to divide instead of finding the area. For Question 2, a student may work out 12 ÷ 6 = 2 because they have just found an acceleration.
| Wrong | Right | |
|---|---|---|
| Question 2 | 12 ÷ 6 = 2 | ½ × 6 × 12 = 36 |
| Unit check | 2 m/s², not metres | 36 m |
| Cause | Gradient used for distance | Area used for distance |
The unit check is the safest catch. If the question asks for metres and your unit is m/s², the method is wrong.
Does the same line always mean the same thing?
No. Draw the line from (0, 0) to (5, 30) and see.
Read as a distance-time graph, the gradient is 30 ÷ 5 = 6 m/s, which is a steady speed.
Read as a speed-time graph, the gradient is 6 m/s², and the area is ½ × 5 × 30 = 75 m.
The picture is the same, but the axis labels decide everything.
Check yourself
A speed-time graph has points (0, 8), (4, 8) and (9, 28). Find the acceleration in the second stage and the distance travelled in the first 4 seconds.
Answer
Acceleration in the second stage is a gradient: (28 − 8) ÷ (9 − 4) = 20 ÷ 5 = 4 m/s².
Distance in the first 4 seconds is an area. The shape is a rectangle: 4 × 8 = 32 m.
Units check: m/s² for the gradient, m for the area.
What to study next
Test the decision on mixed questions in the motion graphs practice set. Use the motion-graph explorer to edit a line and watch gradient and area change, and compare graphs in the graph evidence and fair-comparison lab.
If you want a teacher to check your choices on fresh graphs, see online one-to-one Mathematics tuition.